A General Automatic Method for Optimal Construction of Matrix Product Operators Using Bipartite Graph Theory

Abstract

Constructing matrix product operators (MPO) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For DMRG to be conveniently used in different problems described by different Hamiltonians, in this work we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) It is automatic in that only the definition of the operator is required; (ii) It is symbolic thus free of any numerical error; (iii) The complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; (iv) The symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.

31 pages, 6 figures

References in corpus (8)

• The density-matrix renormalization group in the age of matrix product states
• The Density Matrix Renormalization Group in Chemistry and Molecular Physics: Recent Developments and New Challenges
• Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm
• Generic Construction of Efficient Matrix Product Operators
• Finite automata for caching in matrix product algorithms
• Time-step targeting time-dependent and dynamical density matrix renormalization group algorithms with ab initio Hamiltonians
• Targeted Excited State Algorithms
• Ultrafast ab-initio Quantum Chemistry Using Matrix Product States